7.1.1 1.1

7.1.1.1 [2001] Problem 1

problem number 2001

Added March 23, 2019.

Problem Chapter 1.1.1.1, from Handbook of nonlinear partial differential equations by Andrei D. Polyanin, Valentin F. Zaitsev.

Solve for $w(x,t)$ $$w_t=a{w}_{xx}+b{w}^2$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, t], t] == a*D[w[x, t], {x, 2}] + b*w[x, t]^2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, t], {x, t}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  diff(w(x,t),t)= a*diff(w(x,t),x$2) + b*w(x,t)^2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,t))),output='realtime'));
 

sol=()

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7.1.1.2 [2002] Problem 2

problem number 2002

Added March 23, 2019.

Problem Chapter 1.1.1.2, from Handbook of nonlinear partial differential equations by Andrei D. Polyanin, Valentin F. Zaitsev.

Solve for $w(x,t)$

$$w_t=w_{xx}+aw(1-w)$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, t], t] == D[w[x, t], {x, 2}] + a*w[x, t]*(1 - w[x, t]); 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, t], {x, t}], 60*10]];
 

$$\begin{array}{rl}& \{w(x,t)\to \frac{1}{4}(1+\tanh \left(\frac{1}{12}(5at-\sqrt{6}\sqrt{a}x-12c_3)\right)){}^2\}\\ & \{w(x,t)\to -\frac{1}{4}(-3+\tanh \left(\frac{1}{12}(5at-i\sqrt{6}\sqrt{a}x-12c_3)\right))(1+\tanh \left(\frac{1}{12}(5at-i\sqrt{6}\sqrt{a}x-12c_3)\right))\}\\ & \{w(x,t)\to -\frac{1}{4}(-3+\tanh \left(\frac{1}{12}(5at+i\sqrt{6}\sqrt{a}x-12c_3)\right))(1+\tanh \left(\frac{1}{12}(5at+i\sqrt{6}\sqrt{a}x-12c_3)\right))\}\\ & \{w(x,t)\to \frac{1}{4}(1+\tanh \left(\frac{1}{12}(5at+\sqrt{6}\sqrt{a}x-12c_3)\right)){}^2\}\\ & \{w(x,t)\to \frac{1}{4}(1+\tanh (\frac{5at}{12}-\frac{\sqrt{a}x}{2\sqrt{6}}+c_3)){}^2\}\\ & \{w(x,t)\to -\frac{1}{4}(-3+\tanh (\frac{5at}{12}-\frac{i\sqrt{a}x}{2\sqrt{6}}+c_3))(1+\tanh (\frac{5at}{12}-\frac{i\sqrt{a}x}{2\sqrt{6}}+c_3))\}\\ & \{w(x,t)\to -\frac{1}{4}(-3+\tanh (\frac{5at}{12}+\frac{i\sqrt{a}x}{2\sqrt{6}}+c_3))(1+\tanh (\frac{5at}{12}+\frac{i\sqrt{a}x}{2\sqrt{6}}+c_3))\}\\ & \{w(x,t)\to \frac{1}{4}(1+\tanh (\frac{5at}{12}+\frac{\sqrt{a}x}{2\sqrt{6}}+c_3)){}^2\}\end{array}$$

Maple ✓

restart; 
pde :=  diff(w(x,t),t)= diff(w(x,t),x$2) + a*w(x,t)*(1-w(x,t)); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,t))),output='realtime'));
 

$$w(x,t)=-\frac{({\tanh }^2(-\frac{5at}{12}+c_1+\frac{\sqrt{-6a}x}{12}))}{4}-\frac{\tanh (-\frac{5at}{12}+c_1+\frac{\sqrt{-6a}x}{12})}{2}+\frac{3}{4}$$

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